LIU Ruochuan and Collaborators' Paper Published Online in InventionesMathematicae

Recently, the international top mathematics journal InventionesMathematicae published online the paper "Topological Cyclic Homology of Local Fields" co-authored by Peking University BoyaChair Professor LIU Ruochuan and Shanghai CenterForMathematical Sciences Professor WANG Guozhen.

Topological cyclic homology, an extension of cyclic homology developed by Connes (Fields Medalist, 1983), was introduced by Bökstedt, Xiang Wu-Yi, and Madsen in the 1990s. The work of Dundas, Goodwillie, and McCarthy established topological cyclic homology as an essential tool for computing algebraic K-theory. In 2003, Hesselholt and Madsen first computed the topological cyclic homology of p-adic local fields for all odd prime numbers.

Recently, the work of Bhatt, Morrow, and Scholze revealed a profound connection between topological cyclic homology and p-adiccohomology theories—especially the prismatic cohomology introduced by Bhatt (2022 ICM one-hour speaker, Professor at the Institute for Advanced Study, Princeton) and Scholze (Fields Medalist, 2018). This discovery has made research into topological cyclic homology increasingly significant in arithmetic geometry.

Influenced by these works, LIU Ruochuan and WANG Guozhen collaborated to research topological cyclic homology, proposing a new method for its computation—the descent spectral sequence approach. Using this method, they significantly simplified the computation of topological cyclic homology for p-adic local fields and resolved the previously unsolved case when p="2." Furthermore, the theoretical framework of the descent spectral sequence also provided a stacky reformulation of prismatic cohomology.

The work of LIU Ruochuan and WANG Guozhen has opened new avenues for topological cyclic homology, offering broad application prospects in its computation.

LIU Ruochuan's primary research areas are arithmetic geometry and algebraic number theory. He is an internationally leading mathematician in these fields and has received the inaugural Xplorer Prizeand the second prize of the National Natural Science Award.

WANG Guozhen studied at the School of Mathematical Sciences at Peking University from 2004 to 2011, earning his bachelor's and master's degrees. He is currently a professor at the Shanghai CenterForMathematical Sciences, with a primary research focus on algebraic topology. He was invited to give a talk at the 2022 International Congress of Mathematicians (ICM2022).

Link to Professor LIU Ruochuan's paper: https://link.springer.com/article/10.1007/s00222-022-01134-9